Simplified decoder for a bit interleaved COFDM-MIMO system

ABSTRACT

Multiple input multiple output (MIMO) systems are candidates for higher data rate wireless communication systems. Currently, for a single input single output (SISO) 1802.11a system can provide a transmission data rate up to 54 Mbps. The present invention is a 2 by 2 Multiple In Multiple Out (MIMO) system having a decoding apparatus that increases the data rate to over 100 Mbps but at a computation cost about the same as for optimal decoding for a SISO system and 1/64 of the optimal decoding for a 2 by 2 MIMO system. At a bit error rate (BER) of 10-4 level, the 2 by 2 MIMO system of the present invention provides double the transmission data rate of the SISO system with approximately the same signal to noise ratio (SNR).

CROSS REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. provisional application Ser.No. 60/430,424 filed Dec. 3, 2002, which is incorporated herein byreference.

The present invention relates to a simplified decoder for a codedorthogonal frequency division multiplexing-multiple input multipleoutput (COFDM-MIMO) system. More particularly, the present inventionrelates to a bit interleaved system with maximum (ML) likelihooddecoding. Most particularly the present invention relates to a 2 by 2MIMO system with Zero Forcing (ZF) guided maximum likelihood (ML)decoding that doubles the transmission data rate of a single inputsingle output (SISO) IEEE 802.11a system based on orthogonal frequencydivision multiplexing (OFDM) technique.

MIMO systems have been studied as a promising candidate for the nextgeneration of high data rate wireless communication system. Currently,for a single antenna system (SISO), IEEE 802.11a employing the OFDMmodulation technique has a maximum data transmission rate of 54 Mbps.There is only one transmission antenna and one receiving antenna, i.e.,it is a SISO system, and the signal constellation for 802.11a is 64quadrature amplitude modulation (QAM). Transmission data rates in excessof 100 Mbps is a goal for the next generation wireless communicationsystem.

Given the physical channel characteristics of wireless communicationsystems, it is almost impossible to increase the data rate with a singleantenna system by increasing the order of the constellation of thesignal.

One possible approach to achieving a greater than 100 Mbps data rate isa 2 by 2 MIMO system based on an IEEE 802.11a SISO system in which thetwo transmission antennae transmit different data streams that are codedin the same way as an 802.11a system at each antenna. This system canachieve a transmission data rate of 108 Mbps with approximately the samesignal-to-noise ratio (SNR) as the prior art 54 Mbps IEEE 802.11a SISOsystem based on OFDM modulation that is illustrated in FIG. 1. FIG. 2illustrates a prior art 2 by 2 MIMO system that could be used in thisway.

Suppose the system of FIG. 2 employs optimal decoding and the wirelesschannel is defined as

${H = \begin{pmatrix}h_{11} & h_{21} \\h_{12} & h_{22}\end{pmatrix}},$where h_(ij) 20 represents the channel from transmitter antenna i toreceiver antenna j, i.e., Txi to Rxj. Without losing generality, assumethe four channels are Rayleigh fading channels that are independent ofone another. Then the received signal in frequency domain on subcarrierk can be expressed as

$\begin{matrix}{\begin{pmatrix}r_{1} \\r_{2}\end{pmatrix} = {{\begin{pmatrix}h_{11} & h_{21} \\h_{12} & h_{22}\end{pmatrix}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{1} \\n_{2}\end{pmatrix}}} & (1)\end{matrix}$Since each subcarrier is decoded separately, the subscript ks inequation (1) is omitted. In optimal maximum likelihood (ML) detection,for each received signal pair, r₁ and r₂, to determine whether atransmitted bit in these symbols is ‘1’ or ‘0’, it is necessary to findthe largest probabilitymax(p(r|b))  (2)where

$r = {{\begin{pmatrix}r_{1} \\r_{2}\end{pmatrix}\mspace{14mu}{and}\mspace{14mu} b} = \begin{pmatrix}b_{1i} \\b_{2i}\end{pmatrix}}$are the bits in symbol s₁ and s₂ for which a decision needs to be made.In an add white gaussian noise (AWGN) environment, this is equivalent tofinding

$\begin{matrix}{{\max\limits_{s_{m,}s_{n}}\left( {\left. {\frac{1}{\sqrt{2\;\pi}\sigma}\;{\mathbb{e}}^{\frac{{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2}}{2\sigma^{2}}}*\frac{1}{\sqrt{2\;\pi}\sigma}{\mathbb{e}}^{\frac{{{r_{2} - {h_{21}s_{m}} - {h_{22}s_{n}}}}^{2}}{2\sigma^{2}}}} \middle| b_{1i} \right.,b_{2i}} \right)} = {\max\limits_{s_{m},s_{n}}\left( {\left. {\frac{1}{2\;\pi\;\sigma^{2}}\;{\mathbb{e}}^{\frac{{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2}}{2\sigma^{2}}\frac{{{r_{2} - {h_{21}s_{m}} - {h_{22}s_{n}}}}^{2}}{2\sigma^{2}}}} \middle| b_{1i} \right.,b_{2i}} \right)}} & (3)\end{matrix}$It is also equivalent to finding

$\begin{matrix}{\min\limits_{s_{m},s_{n}}\left( {\left. {{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{21}s_{m}} - {h_{22}s_{n}}}}^{2}} \middle| b_{1i} \right.,b_{2i}} \right)} & (4)\end{matrix}$In order to determine the bit metrics for a bit in symbol s₁, thefollowing equation must be evaluated. For bit i in symbol s₁ to be ‘0’,it is necessary to evaluate

$\begin{matrix}{m_{1i}^{0} = {\min\limits_{{s_{m} \in S^{0}},{s_{n} \in S}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{1i} \right. = 0} \right)}} & (5)\end{matrix}$Where m_(1i) ⁰ represents the bit metrics for bit i in received symbols₁ to be ‘0’. S represents for the whole constellation point set, whileS⁰ represents the subset of the constellation point set such that bitb_(i)=0. For bit i in symbol s₁ to be ‘1’, it is necessary to evaluate

$\begin{matrix}{m_{1i}^{1} = {\min\limits_{{s_{m} \in S^{1}},{s_{n} \in S}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{1i} \right. = 1} \right)}} & (6)\end{matrix}$where S¹ represents the subset of the constellation point set such thatbit b_(i)=1.

Using the same method, it is possible to determine the bit metrics fortransmitted symbol s₂. For bit i in symbol s₂ to be ‘0’, it is necessaryto evaluate

$\begin{matrix}{m_{2i}^{0} = {\min\limits_{{s_{m} \in S},{s_{n} \in S^{0}}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{2i} \right. = 0} \right)}} & (7)\end{matrix}$For bit i in symbol s₂ to be ‘1’, it is necessary to evaluate

$\begin{matrix}{m_{2i}^{1} = {\min\limits_{{s_{m} \in S},{s_{n} \in S^{1}}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{2i} \right. = 1} \right)}} & (8)\end{matrix}$Then, the bit metrics pairs (m_(1i) ⁰, m_(1i) ¹) (m_(2i) ⁰, m_(2i) ¹)are sent to corresponding deinterleavers and Viterbi decoders for FECdecoding of each of the data streams.

Simulation results show that using optimal decoding, the proposed 108Mbps MIMO system actually performs 4 dB better than the SISO 54 Mbpssystem at a BER of 10⁻⁴. However, the computation cost for the optimaldecoding is very high. To obtain bit metrics for a bit in signal s₁ tobe 0 and 1, it is necessary to evaluate 64*64 permutations of the s₁ ands₂ constellation, which cannot be accomplished cost effectively withexisting computational capabilities. The computation cost for this 2 by2 MIMO system decoding is too high to be practical.

Thus, there is a need for an alternative coding method to reduce thehigh computation cost when a 2 by 2 MIMO system based on and 54 MbpsIEEE 802.11a SISO system is employed for increasing the datatransmission rate above 100 Mbps.

The present invention is a 108 Mbps 2 by 2 MIMO system based on a 54Mbps SISO system, as illustrated in FIG. 3, that replaces optimaldecoding with a simplified decoding method that has about the samecomputation cost as the optimal SISO decoder and about 1/64 thecomputation cost of the optimal MIMO decoder. In the system illustratedin FIG. 3, the separate demapping an deinterleaving module 10, of theprior art system illustrated in FIG. 1, is replaced by a shareddemapping and signal separation unit 34 and the separate deinterleavingunits 30 and 31.

The present invention employs a ZF guided maximum likelihood (ML)decoding method. For a SISO single carrier system, since atime-dispersed channel (frequency selective fading channel) brings thechannel memory into the system, joint maximum likelihood (ML)equalization and decoding is not realistic because of the highcomputation cost. The general practice is to first useminimum-mean-square-error/zero forcing (MMSE/ZF) as the criteria toequalize the channel. Then the equalized signal is sent to a maximumlikelihood (ML) detector for further decoding. However, this is asub-optimal system.

In a SISO OFDM system, since the system is designed to let eachsub-carrier experience flat fading channel, the real maximum likelihood(ML) equalization and decoding can be implemented with affordablecomputational cost. Yet in a MIMO OFDM system, because of the largenumber of permutation evaluations of the constellation set required inthe metrics calculation, the computation cost for real maximumlikelihood (ML) equalization and decoding is too high to be practical.

One way to avoid the large number of permutation computations is tofirst find the approximate value of the transmitted symbols s₁ and s₂and then use the maximum likelihood (ML) detection method to find thebit metrics for s₁ while taking s₂ as the value calculated by the ZFmethod. It is reasonable to make the assumption that when the SNR ishigh enough, the ZF decision is very close to the optimal maximumlikelihood decision. Thus, the present invention incurs approximatelythe same computation cost in a MIMO system to get the bit metrics forthe transmitted symbols s₁ and s₂ as the SISO system incurs fortransmitted symbols s.

FIG. 1 illustrates a prior art 54 Mbps IEEE 802.11a SISO system based onOFDM modulation.

FIG. 2 illustrates a prior art 2 by 2 MIMO system.

FIG. 3 illustrates a 108 Mbps 2 by 2 MIMO system based on the 54 MbpsSISO system of FIG. 1, according to a preferred embodiment of thepresent invention.

FIGS. 4A-C illustrate a Slice-Compare-Selection Operation.

FIG. 5 shows simulation results comparing the 108 Mbps MIMO system ofFIG. 3 with the 54 Mbps SISO system of FIG. 1.

The preferred embodiments of the present invention employ a simplifieddecoding method. The details of the simplified decoding method aredescribed below with reference to the drawings.

The received signal can be written as

$\begin{pmatrix}r_{1} \\r_{2}\end{pmatrix} = {{\begin{pmatrix}h_{11} & h_{21} \\h_{12} & h_{22}\end{pmatrix}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}} + {\begin{pmatrix}n_{1} \\n_{2}\end{pmatrix}.}}$According to the ZF criteria, the transmitted signal can be estimated bythe demapping and signal separation module 34 as

$\begin{matrix}{\begin{pmatrix}{\overset{\sim}{s}}_{1} \\{\overset{\sim}{s}}_{2}\end{pmatrix} = {\begin{pmatrix}h_{11} & h_{21} \\h_{12} & h_{22}\end{pmatrix}^{- 1}\begin{pmatrix}r_{1} \\r_{2}\end{pmatrix}}} & (9)\end{matrix}$

Using the minimum Euclidean distance calculated for the ZF calculatedsymbol and constellation point, the demapping and signal separationmodule 34 obtains the estimated transmitted symbol by hard decision. Thesymbols after the hard decision operation can be represented as

$\begin{pmatrix}{\hat{s}}_{1} \\{\hat{s}}_{2}\end{pmatrix}.$The bit metrics for transmitted symbol s₁ are then calculated by thedemapping and signal separation module 34 as

$\begin{matrix}{m_{1i}^{0} = {\min\limits_{s_{m} \in S^{0}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}{\hat{s}}_{2}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}{\hat{s}}_{2}}}}^{2}} \right) \middle| b_{1i} \right. = 0} \right)}} & (10) \\{m_{1i}^{1} = {\min\limits_{s_{m} \in S^{1}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}{\hat{s}}_{2}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}{\hat{s}}_{2}}}}^{2}} \right) \middle| b_{1i} \right. = 1} \right)}} & \;\end{matrix}$and bit metrics for transmitted symbol s₂ can then be calculated as

$\begin{matrix}{m_{2i}^{0} = {\min\limits_{s_{n} \in S^{0}}\left( {\left. \left( {{{r_{1} - {h_{11}{\hat{s}}_{1}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}{\hat{s}}_{1}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{2i} \right. = 0} \right)}} & (11) \\{m_{2i}^{1} = {\min\limits_{s_{n} \in S^{1}}\left( {\left. \left( {{{r_{1} - {h_{11}{\hat{s}}_{1}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}{\hat{s}}_{1}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{2i} \right. = 1} \right)}} & \;\end{matrix}$where S^(p) represents the subset of the constellation points such thatbit b_(i) is p where p=0 or 1. Then, the bit metrics pairs (m_(1i) ⁰,m_(1i) ¹) (m_(2i) ⁰, m_(2i) ¹) are sent to corresponding first andsecond deinterleavers 30 and 31 and different Viterbi decoders 33 and34, respectively, for forward error correction (FEC) decoding of eachdata stream.

In a second preferred embodiment, a further simplified decoding methodis provided based on the first preferred embodiment. Unlike the firstpreferred embodiment in which the demapping and signal separation module34 uses the MIMO ML criteria to calculate the bit metrics for each bitin the two transmitted symbols after the ZF operation, the SISO ML isused by the demapping and signal separation module 34 to find theconstellation points for each bit that satisfy

$\begin{matrix}{\min\limits_{s \in S_{i}^{p}}{{{\overset{\sim}{s}}_{q} - s}}^{2}} & (12)\end{matrix}$where q=1,2 and pε{0,1}. Two constellation points are defined by thedemapping and signal separation module 34 that correspond to the bitmetrics calculation of (12) for bit i of the transmitted symbol s_(q) tobe s_(qi) ^(p). In SISO decoding, bit metrics calculated from (12) aresent to a Viterbi decoder for decoding. In MIMO decoding, equation (12)is only used by the demapping and signal separation module 34 todetermine the constellation points that satisfy (12) and use theseconstellation points in MIMO ML criteria to calculate the bit metricsfor each bit that are sent to a Viterbi decoder for decoding. That is,the bit metrics are calculated by the demapping and signal separationmodule 34 asm _(1i) ^(p)=(∥r ₁ −h ₁₁ s _(1i) ^(p) −h ₂₁ ŝ ₂∥² +∥r ₂ −h ₁₂ s _(1i)^(p) −h ₂₂ ŝ ₂∥²)m _(2i) ^(p)=(∥r ₁ −h ₁₁ ŝ ₁ −h ₂₁ s _(2i) ^(p)∥² +∥r ₂ −h ₁₂ ŝ ₁ −h ₂₂s _(2i) ^(p)∥²)  (13)

Then, the bit metrics pairs (m_(1i) ⁰, m_(1i) ¹) (m_(2i) ⁰, m_(2i) ¹)are sent to corresponding first and second deinterleavers 30 and 31 anddifferent Viterbi decoders 33 and 34, respectively, for forward errorcorrection (FEC) decoding of each data stream.

In a hardware implementation, the 12 constellation points for the 6 bitsin one transmitted symbol can be obtained by a slice-compare-selectoperation. An example of quadrature-phase shift keying (QPSK) isillustrated in FIG. 4A. If the real part of the received symbol isconsidered, it is possible to determine that the two constellationpoints corresponding to bit b₀ are the two points connected by thedashed line in FIG. 4B. The same method can be used to determine thecorresponding constellation points for bit b₁ by using the imaginarypart of the received symbol, as shown in FIG. 4C. With the slicingmethod, the actual distance calculation of equation (12) is not needed.In the second preferred embodiment, the permutation in distancesearching in the MIMO ML bit metrics calculation can be avoided, whichreduces the computation cost of the MIMO ML bit metrics calculation.

Simulation results, shown in FIG. 5, confirm the performance of bothembodiments of the present invention. The multipath channel simulated isthe exponential Rayleigh fading channel defined in Bob O'Hara, AlPetrick; “The IEEE 802.11 Handbook: A Designer's Companion”, December1999, having a 40 ns rms delay spread. The four channels across the twotransmission antennae and two receiving antennae are independent of eachother, which means there is no correlation between any of the fourchannels. For each data point of the sign-to-noise-ratio vs.bit-error-rate (SNR vs. BER) curves of FIG. 5, 1 million bits equallydistributed in 250 packets was simulated. It is reasonable to assumethat the wireless channel for each antenna element is the same for eachpacket, while it is different for different packets. In all thesimulations, ideal frequency and timing synchronization is assumed.

Simulation results show that although the performance of the firstembodiment of the simplified decoding method of the present invention isabout 4 dB worse than the optimal decoding method at a BER level of10⁻⁴, it is almost the same as the optimal decoding for the SISO systemat 54 Mbps 43. This result shows that the first embodiment of thepresent invention comprising a 2 by 2 MIMO system 41 can double thetransmission data rate of the SISO system 43 for the same SNR atreasonable computation cost. The second embodiment provides the sameimprovement for a further reduced computation cost. Therefore, thesimulation show that both embodiments of the present invention haveabout the same BER vs SNR performance, which is almost the same as SISO54 Mbps system 43 and 4 dB less than MIMO optimal decoding system 42 atBER level of 10⁻⁴. And, the increase in transmission rate by double isobtained for no increase in computation cost in the first embodiment anda reduced computation cost in the second embodiment

Referring to FIG. 4, a 2 by 2 MIMO system 42 based on IEEE 802.11a SISOsystem according to the present invention can provide a 108 Mbpstransmission data rate that doubles the data rate of the IEEE 802.11aSISO system within the same range of SNR. Optimal decoding of the MIMOsystem 42 according to the prior art provides 4 dB better BER vs. SNRperformance than the SISO 54 Mbps system 43 at BER level of 10⁻⁴ but thehigh computation cost of the optimal decoding makes such animplementation impractical. The present invention provides two preferredembodiments for ZF guided simplified MIMO decoding, 40 and 41, havingcomputation costs that are almost the same as that of the optimaldecoder for the 54 Mbps SISO system 43. Although each of the embodimentsfor a simplified method, 40 and 41, performs 4 dB worse than the optimaldecoder for MIMO system 42, each provides almost the same SNRperformance as the SISO 54 Mbps system 43 at the BER level of 10⁻⁴, butat the transmission data rate of 108 Mbps While the examples providedillustrate and describe a preferred embodiment of the present invention,it will be understood by those skilled in the art that various changesand modifications may be made, and equivalents may be substituted forelements thereof without departing from the true scope of the presentinvention. In addition, many modifications may be made to adapt theteaching of the present invention to a particular situation withoutdeparting from the central scope. Therefore, it is intended that thepresent invention not be limited to the particular embodiments disclosedas the best mode contemplated for carrying out the present invention,but that the present invention include all embodiments falling withinthe scope of the appended claims.

1. A 2×2 wireless local area network, comprising: a first and a secondsingle input single output (SISO) system respectively having a first andsecond transmitter antenna that transmit a first and second transmittedsignal s₁ and s₂; a first and second receiver antenna that receive afirst and second received signal r₁ and r₂; and a demapping and signalseparation module that employs zero forcing (ZF) to guide maximumlikelihood (ML) decoding and is connected to said first and second SISOsystem and is adapted to process said first and second received signal,wherein a data transmission rate of the 2×2 system is greater than 100Mbps at a bit error rate of 10⁻⁴ and a computation cost for decoding onthe order of the decoding cost for an optimal SISO system; wherein thedemapping and signal separation module employs zero forcing (ZF) of thefirst and second received signal, respectively, which received signalscorrespond to $\begin{pmatrix}r_{1} \\r_{2}\end{pmatrix} = {{\begin{pmatrix}h_{11} & h_{21} \\h_{12} & h_{22}\end{pmatrix}\begin{pmatrix}s_{1} \\s_{2}\end{pmatrix}} + \begin{pmatrix}n_{1} \\n_{2}\end{pmatrix}}$  and results in the first and second transmitted signalbeing calculated as $\begin{pmatrix}{\overset{\sim}{s}}_{1} \\{\overset{\sim}{s}}_{2}\end{pmatrix} = {\begin{pmatrix}h_{11} & h_{21} \\h_{12} & h_{22}\end{pmatrix}^{- 1}\begin{pmatrix}r_{1} \\r_{2}\end{pmatrix}}$  and finds the minimum Euclidean distance between ZFcalculated symbol and a constellation point to estimate by hard decisionthe first and second transmitted signal as first and second estimatedsignal as $\begin{pmatrix}{\hat{s}}_{1} \\{\hat{s}}_{2}\end{pmatrix}\quad$  which are then used to guide the maximum likelihood(ML) decoding, wherein, h_(ij) represents a channel from the i^(th)transmitter antenna to the j^(th) receiver antenna and n_(i) are noisesignal for i,j=1,2.
 2. The 2×2 system of claim 1, wherein respective bitmetrics calculated for first and second transmitted signal s₁ and s₂ arecalculated using $\begin{pmatrix}{\hat{s}}_{1} \\{\hat{s}}_{2}\end{pmatrix}\quad$ as follows for s₁ $\begin{matrix}{m_{1i}^{0} = {\min\limits_{s_{m} \in S^{0}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}{\hat{s}}_{2}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}{\hat{s}}_{2}}}}^{2}} \right) \middle| b_{1i} \right. = 0} \right)}} \\{m_{1i}^{1} = {\min\limits_{s_{m} \in S^{1}}\left( {\left. \left( {{{r_{1} - {h_{11}s_{m}} - {h_{21}{\hat{s}}_{2}}}}^{2} + {{r_{2} - {h_{12}s_{m}} - {h_{22}{\hat{s}}_{2}}}}^{2}} \right) \middle| b_{1i} \right. = 1} \right)}}\end{matrix}$ and for s₂ $\begin{matrix}{m_{2i}^{0} = {\min\limits_{s_{n} \in S^{0}}\left( {\left. \left( {{{r_{1} - {h_{11}{\hat{s}}_{1}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}{\hat{s}}_{1}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{2i} \right. = 0} \right)}} \\{m_{2i}^{1} = {\min\limits_{s_{n} \in S^{1}}\left( {\left. \left( {{{r_{1} - {h_{11}{\hat{s}}_{1}} - {h_{21}s_{n}}}}^{2} + {{r_{2} - {h_{12}{\hat{s}}_{1}} - {h_{22}s_{n}}}}^{2}} \right) \middle| b_{2i} \right. = 1} \right)}}\end{matrix}$ and the bit metrics pairs (m_(1i) ⁰, m_(1i) ¹) (m_(2i) ⁰,m_(2i) ¹) are sent to a respective first and second deinterleaver and afirst and secondViterbi decoder for decoding, wherein b_(1i) and b_(2i)respectively is a bit in signal s₁ and s₂ for which a decision is beingmade.
 3. The 2×2 system of claim 1, wherein the demapping and signalseparation module obtains a first and second constellation point, s_(1i)^(p) and s_(2i) ^(p), satisfying the minimum Euclidean distance fromfirst and second transmitted signal {tilde over (s)}₁ and {tilde over(s)}₂ for bit i$\min\limits_{s \in S_{i}^{p}}{{{\overset{\sim}{s}}_{q} - s}}^{2}$where q=1, 2 and s_(i) ^(p) represents the subset of a constellationpoint set for bit i, the bit for which a decision is being made, suchthat pε{0,1}, uses these constellation points as input to a maximumlikelihood calculation in the form of a bit metrics calculation forp=0,1 $\begin{matrix}{m_{1\; i}^{p} = \left( {{{{r_{1} - {h_{11}s_{1i}^{p}} - {h_{21}{\hat{s}}_{2}}}}}^{2} + {{{r_{2} - {h_{12}s_{1i}^{p}} - {h_{22}{\hat{s}}_{2}}}}}^{2}} \right)} \\{m_{2i}^{p} = \left( {{{{r_{1} - {h_{11}{\hat{s}}_{1}} - {h_{21}s_{2i}^{p}}}}}^{2} + {{{r_{2} - {h_{12}{\hat{s}}_{1}} - {h_{22}s_{2i}^{p}}}}}^{2}} \right)}\end{matrix}$ and the bit metrics pairs (m_(1i) ⁰, m_(1i) ¹) (m_(2i) ⁰,m_(2i) ¹) are sent to a respective first and second deinterleaver and afirst and second Viterbi decoder for decoding.
 4. The 2×2 system ofclaim 1, wherein the demapping and signal separation module performs aslice-compare-select operation to determine a first and secondconstellation point, s_(1i) ^(p) and s_(2i) ^(p), corresponding to theZF signal which first and second constellation point is used as input toa maximum likelihood calculation in the form of a bit metricscalculation for p=0,1 $\begin{matrix}{m_{1\; i}^{p} = \left( {{{{r_{1} - {h_{11}s_{1i}^{p}} - {h_{21}{\hat{s}}_{2}}}}}^{2} + {{{r_{2} - {h_{12}s_{1i}^{p}} - {h_{22}{\hat{s}}_{2}}}}}^{2}} \right)} \\{m_{2i}^{p} = \left( {{{{r_{1} - {h_{11}{\hat{s}}_{1}} - {h_{21}s_{2i}^{p}}}}}^{2} + {{{r_{2} - {h_{12}{\hat{s}}_{1}} - {h_{22}s_{2i}^{p}}}}}^{2}} \right)}\end{matrix}$ and the bit metrics pairs (m_(1i) ⁰, m^(1i) ¹) (m_(2i) ⁰,m_(2i) ¹) are sent to a respective first and second deinterleaver and afirst and second Viterbi decoder for decoding.
 5. The 2×2 system ofclaim 1, wherein said first and second SISO system is based on a 54 MbpsIEEE 802.11a SISO orthogonal frequency division multiplexing (OFDM)system.